Understanding the Ratio of Radii of Two Spheres Given Their Surface Areas

When working with spheres, one common problem involves finding the ratio of the radii given the surface areas. This article will explore how to solve such a problem step by step, providing a clear understanding of the underlying concepts.

Conceptual Background

The surface area of a sphere is given by the formula A 4πr2, where π is the mathematical constant pi (approximately 3.14159), and r is the radius of the sphere. Understanding this formula is crucial to solving problems related to the relationship between the surface areas of different spheres.

Problem Statement

Given the surface areas of two spheres, 25 cm2 and 30 cm2, we need to find the ratio of their radii.

Solving the Problem

Let r1 and r2 be the radii of the two spheres. Given:

Surface area of sphere 1 4πr12 25 cm2

Surface area of sphere 2 4πr22 30 cm2

Step-by-Step Solution

Start with the formula for the surface area of a sphere:

( frac{4pi r_{1}^{2}}{4pi r_{2}^{2}} frac{25}{30} )

Cancel out the 4π from both the numerator and the denominator:

( frac{r_{1}^{2}}{r_{2}^{2}} frac{25}{30} )

Simplify the fraction:

( frac{r_{1}^{2}}{r_{2}^{2}} frac{5}{6} )

Take the square root of both sides to solve for the ratio of the radii:

( frac{r_{1}}{r_{2}} sqrt{frac{5}{6}} )

Thus, the ratio of the radii is:

( frac{r_{1}}{r_{2}} sqrt{frac{5}{6}} )

Express the ratio in a more conventional form:

( r_{1} : r_{2} sqrt{5} : sqrt{6} )

Conclusion

By using the surface area formula and the properties of ratios, we can easily find the ratio of the radii of two spheres given their surface areas. This problem demonstrates the relationship between the surface areas and the radii of spheres, which is a fundamental concept in geometry.

Additional Insights

The ratio of the areas of two figures is equal to the square of the ratio of their corresponding linear dimensions. This principle is widely applicable in geometry and helps in solving a variety of problems related to the properties of shapes and figures.

Understanding the ratio of the radii given their surface areas can be particularly useful in various fields, including physics, engineering, and architecture, where the properties of spheres and other 3D shapes are crucial.